Elasticity (physics)

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Elasticity" physics – news · newspapers · books · scholar · JSTOR (February 2017) (Learn how and when to remove this template message)

Part of a series on
Continuum mechanics
${\displaystyle J=-D{\frac {d\varphi }{dx}}}$ Fick's laws of diffusion
Laws Conservations Mass Momentum Energy Inequalities Clausius–Duhem (entropy)
Solid mechanics Deformation Elasticity linear Plasticity Hooke's law Stress Strain Finite strain Infinitesimal strain Compatibility Bending Contact mechanics frictional Material failure theory Fracture mechanics
Fluid mechanics Fluids Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations Poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure Liquids Adhesion Capillary action Chromatography Cohesion (chemistry) Surface tension Gases Atmosphere Boyle's law Charles's law Combined gas law Fick's law Gay-Lussac's law Graham's law Plasma
Rheology Viscoelasticity Rheometry Rheometer Smart fluids Electrorheological Magnetorheological Ferrofluids
Scientists Bernoulli Boyle Cauchy Charles Euler Fick Gay-Lussac Graham Hooke Newton Navier Noll Pascal Stokes Truesdell
v t e

In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.

The physical reasons for elastic behavior can be quite different for different materials. In metals, the atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied.

In

When an elastic material is deformed due to an external force, it experiences internal resistance to the deformation and restores it to its original state if the external force is no longer applied. There are various elastic moduli, such as Young's modulus, the shear modulus, and the bulk modulus, all of which are measures of the inherent elastic properties of a material as a resistance to deformation under an applied load. The various moduli apply to different kinds of deformation. For instance, Young's modulus applies to extension/compression of a body, whereas the shear modulus applies to its shear.^[1] Young's modulus and shear modulus are only for solids, whereas the bulk modulus is for solids, liquids, and gases.

The elasticity of materials is described by a

The SI unit for elasticity and the elastic modulus is the pascal (Pa). This unit is defined as force per unit area, generally a measurement of pressure, which in mechanics corresponds to stress. The pascal and therefore elasticity have the dimension L⁻¹⋅M⋅T⁻².

For most commonly used engineering materials, the elastic modulus is on the scale of gigapascals (GPa, 10⁹ Pa).

Linear elasticity

As noted above, for small deformations, most elastic materials such as

${\displaystyle x}$

${\displaystyle F=kx,}$

where k is a constant known as the rate or spring constant. It can also be stated as a relationship between

${\displaystyle \sigma }$

${\displaystyle \varepsilon }$

${\displaystyle \sigma =E\varepsilon ,}$

where E is known as the Young's modulus.^[7]

Although the general proportionality constant between stress and strain in three dimensions is a 4th-order tensor called stiffness, systems that exhibit symmetry, such as a one-dimensional rod, can often be reduced to applications of Hooke's law.

Finite elasticity

The elastic behavior of objects that undergo finite deformations has been described using a number of models, such as

A material is said to be Cauchy-elastic if the

${\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})}$

It is generally incorrect to state that Cauchy stress is a function of merely a

Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses might depend on the path of deformation. Therefore, Cauchy elasticity includes non-conservative "non-hyperelastic" models (in which work of deformation is path dependent) as well as conservative "hyperelastic material" models (for which stress can be derived from a scalar "elastic potential" function).

Hypoelastic materials

A hypoelastic material can be rigorously defined as one that is modeled using a constitutive equation satisfying the following two criteria:^[8]

1. The Cauchy stress ${\displaystyle {\boldsymbol {\sigma }}}$ at time ${\displaystyle t}$ depends only on the order in which the body has occupied its past configurations, but not on the time rate at which these past configurations were traversed. As a special case, this criterion includes a Cauchy elastic material, for which the current stress depends only on the current configuration rather than the history of past configurations.
2. There is a tensor-valued function ${\displaystyle G}$ such that ${\displaystyle {\dot {\boldsymbol {\sigma }}}=G({\boldsymbol {\sigma }},{\boldsymbol {L}})\,,}$ in which ${\displaystyle {\dot {\boldsymbol {\sigma }}}}$ is the material rate of the Cauchy stress tensor, and ${\displaystyle {\boldsymbol {L}}}$ is the spatial
   velocity gradient
   tensor.

If only these two original criteria are used to define hypoelasticity, then

Note that the second criterion requires only that the function ${\displaystyle G}$ exists. As detailed in the main hypoelastic material article, specific formulations of hypoelastic models typically employ so-called objective rates so that the ${\displaystyle G}$ function exists only implicitly and is typically needed explicitly only for numerical stress updates performed via direct integration of the actual (not objective) stress rate.

Hyperelastic materials

Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a

${\displaystyle {\boldsymbol {\sigma }}={\cfrac {1}{J}}~{\cfrac {\partial W}{\partial {\boldsymbol {F}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}$

This formulation takes the energy potential (W) as a function of the

${\displaystyle {\boldsymbol {F}}}$

${\displaystyle {\boldsymbol {C}}:={\boldsymbol {F}}^{\textsf {T}}{\boldsymbol {F}}}$

${\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}~{\boldsymbol {F}}{\cfrac {\partial W}{\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}$

Applications

Linear elasticity is used widely in the design and analysis of structures such as

Hyperelasticity is primarily used to determine the response of elastomer-based objects such as gaskets and of biological materials such as soft tissues and cell membranes.

Factors affecting elasticity

In a given

Notes

References